Decision and Separability Problems for Baumslag-Solitar Semigroups

We show that the semigroups Sk,` having semigroup presentations 〈a, b : abk = b`a〉 are residually finite and finitely separable. Generally, these semigroups have finite separating images which are finite groups and other finite separating images which are semigroups of order-increasing transformations on a finite partially ordered set. These semigroups thus have vastly different residual and separability properties than the Baumslag-Solitar groups which contain them. 0: Introduction and Notation Let k and ` be nonnegative integers. Throughout, we will write Sk,` for the semigroup having semigroup presentation 〈a, b : ab = ba 〉. We regard these semigroups as being, in some respects, analogous to the groups Gk,` having group presentations Gp〈a, b : ab = ba 〉. The groups Gk,` are generally referred to as Baumslag-Solitar groups. Since the Baumslag-Solitar groups have one-relator presentations, it has long been known, by a theorem of Magnus, [15], that they have solvable word problems. The Baumslag-Solitar groups are HNN-extensions over the integers, so brief modern proofs that they have solvable word problems follow from Britton’s Lemma. By a theorem of Baumslag and Solitar, [3], many of the Baumslag-Solitar groups are non-hopfian and hence are not residually finite [16]. For a more recent account of Baumslag-Solitar groups, see also [2]. When k ≥ 1 and ` ≥ 1, it is known by a theorem of Adjan, [1, Section II, Theorem 3], that the natural homomorphism from Sk,` to Gk,` is an embedding, and hence the presentations given for Sk,` also have solvable word problems. Moreover, [1, Section II, Theorem 1], a left-cancellation law holds in Sk,` provided that ` ≥ 1 and a right-cancellation law holds in Sk,` provided that k ≥ 1. In Section 1, we will give direct explicit solutions of the word problem for the semigroups Sk,` (even in the fairly simple cases where k and/or ` are 0). Later, we will outline an alternate proof of the cancellation laws. When A is a set, regarded as an alphabet of letters, we will use A to denote the free semigroup of nonempty words on A. The free monoid, A∗ on A is the set of all words on A, including the empty word. We will write 1 for the empty word. 1991 Mathematics Subject Classification. 20M05. The author thanks the Mathematics Department at the University of Illinois at UrbanaChampaign for its hospitality while much of this research was done. Especial thanks are owed to Professors Paul Schupp and Derek Robinson for helpful conversations.